Find
\[\cos \left( 6 \arccos \frac{1}{3} \right).\]
Solution: Let $x = \arccos \frac{1}{3},$ so $\cos x = \frac{1}{3}.$  From the triple angle formula,
\[\cos 3x = 4 \cos^3 x - 3 \cos x = 4 \left( \frac{1}{3} \right)^3 - 3 \cdot \frac{1}{3} = -\frac{23}{27}.\]Then from the double angle formula,
\[\cos 6x = 2 \cos^2 3x - 1 = 2 \left( -\frac{23}{27} \right)^2 - 1 = \boxed{\frac{329}{729}}.\]